Comparison Between Effective and True Parameters

The parameters obtained by fitting the observed energy levels to the eigenvalues of an effective Hamiltonian are not unique. Their values depend on (1) the precision and completeness of the input data (2) the size of the matrix that is actually diagonalized (3) the choice of model Hamiltonian and (4) choices such as which parameters to vary, which parameters will be held fixed at values different from zero, and the imposition of constraints on varied parameters. Spectroscopists seldom agree on the meaning of the term deperturbed or on what information belongs in Heff. 

This book was written to help spectroscopists understand the relationship between the exact molecular Hamiltonian, effective Hamiltonians used in fitting spectral data, and the molecular parameters obtained from both spectra and ab initio calculations. Although the general ideas for constructing effective Hamiltonians (Section 4.2) and several examples appropriate to special cases (for example the 2E+ 2n interaction in Sections 3.5.4 and 5.5) are discussed, no attempt is made here to present a complete and universal effective Hamiltonian for diatomic molecules. Brown, et al., (1979) derive an effective Hamiltonian that should be the starting point for the fitting of most non-1E, perturbation-free, diatomic molecular spectra. Other less general, effective Hamiltonians have been proposed, by De Santis, et al., (1973) for 3E states, by Brown and Milton (1976) for S E-states, and by Brown and Merer (1979) for S 1 n-states. 

The idea of an effective Hamiltonian for diatomic molecules was first articulated by Tinkham and Strandberg (1955) and later developed by Miller (1969) and Brown, et al., (1979). The crucial idea is that a spectrum-fitting model (for example Eq. 18 of Brown, et al., 1979) be defined in terms of the minimum number of linearly independent fit parameters. These fit parameters have no physical significance. However, if they are defined in terms of sums of matrix elements of the exact Hamiltonian (see Tables I and II of Brown, et al., 1979) or sums of parameters appropriate to a special limiting case (such as the unique perturber approximation, see Table III of Brown, et al., 1979, or pure precession, Section 5.5), then physically significant parameters suitable for comparison with the results of ab initio calculations are usually derivable from fit parameters. 

The following example illustrates the critical dependence of the values of the fit parameters on the dimension of the effective Hamiltonian matrix. 

However, Eqs. (4.4.39a) - (4.4.40)) can give erroneous results. Gallusser and Dressier (1982) have performed a simultaneous, multistate deperturbation on the NO B2n (v = 0—37) and L2n (v = 0—11) valence states and C2n (v = 0—9), K2n (v = 0 — 4), and Q2n (v = 0 — 3) Rydberg states (see Section 6.2.2). They diagonalized 69 x 69 matrices rather than Van Vleck transform-corrected 2x2 matrices and found that in some cases the third-order correction to.,  

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